4.16.27 \(a (x-y(x))+y'(x)^2-2 y'(x)=0\)

ODE
\[ a (x-y(x))+y'(x)^2-2 y'(x)=0 \] ODE Classification

[[_homogeneous, `class C`], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.28003 (sec), leaf count = 76

\[\left \{\left \{y(x)\to \frac {1}{4} \left (a \left (-2 \sqrt {2} c_1 x+2 c_1^2+x^2\right )-\frac {4}{a}+4 x\right )\right \},\left \{y(x)\to \frac {1}{4} a \left (2 \sqrt {2} c_1 x+2 c_1^2+x^2\right )-\frac {1}{a}+x\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 39

\[ \left \{ y \relax (x ) ={\frac {ax-1}{a}},[x \left ({\it \_T} \right ) =2\,{\frac {{\it \_T}}{a}}+{\it \_C1},y \left ({\it \_T} \right ) ={\frac {{\it \_C1}\,a+{{\it \_T}}^{2}}{a}}] \right \} \] Mathematica raw input

DSolve[a*(x - y[x]) - 2*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-4/a + 4*x + a*(x^2 - 2*Sqrt[2]*x*C[1] + 2*C[1]^2))/4}, {y[x] -> -a^(
-1) + x + (a*(x^2 + 2*Sqrt[2]*x*C[1] + 2*C[1]^2))/4}}

Maple raw input

dsolve(diff(y(x),x)^2-2*diff(y(x),x)+a*(x-y(x)) = 0, y(x),'implicit')

Maple raw output

y(x) = (a*x-1)/a, [x(_T) = 2*_T/a+_C1, y(_T) = (_C1*a+_T^2)/a]